Timeline for Lifting $\mathbb C^*$ actions to holomorphic bundles
Current License: CC BY-SA 3.0
8 events
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| Aug 3, 2017 at 12:34 | comment | added | Jason Starr | This is from long ago, but I just remembered now. I did try to find such a counterexample at the time, but with no success. Now I believe (vaguely) that there are no such counterexamples. | |
| May 12, 2017 at 18:42 | comment | added | aglearner | Jason, thank you for the idea. In order to model at least some bits of what you suggest, could one take $\mathbb CP^3$ with a $\mathbb C^*$-action that fixes a $\mathbb CP^2$ and a point and blow $\mathbb CP^3$ along a smooth cubic $E\subset \mathbb CP^2$? Then there is a rigid $E$ in the blown up variety. But I don't know how to calculate the cohomology groups. Also I have a question on rigidity. Is the bundle $O(-1)\oplus O(-1)$ on $\mathbb CP^1$ considered to be rigid? (I know it can be deformed to $O\oplus O(-2)$) | |
| May 12, 2017 at 15:37 | comment | added | Jason Starr | If I were trying to find a counterexample, I would begin with a threefold $X$ and an elliptic curve $E$ embedded in $X$. If $X$ is simply connected, then by Serre's correspondence, $E$ is the zero scheme of a regular section of a rank $2$ locally free sheaf $V$. If $h^1(X,V)$ is zero and if $E$ is rigid, then $V$ is rigid. If $h^0(X,V)$ equals $1$, then $V$ has no compatible $\mathbb{C}^*$-action. | |
| May 11, 2017 at 4:08 | history | edited | aglearner | CC BY-SA 3.0 |
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| May 11, 2017 at 3:53 | comment | added | aglearner | Allen thank you for this comment, I am a bit worried indeed of this type of counter-examples. Though I see them as of "topological nature". If by any chance there is such a topological obstruction for lifting $\mathbb C^*$ actions, I would also ask if one can lift instead a $\mathbb C^*$-action induced by a finite cover $\mu_n: \mathbb C^*\to \mathbb C^*$ for some $n$. | |
| May 11, 2017 at 1:26 | comment | added | Allen Knutson | Certainly the answer is no if the group is nonabelian: consider $PSL_2$ acting on $\mathbb P^1$, an action that doesn't extend to $\mathcal O(1)$, since $PSL_2$ couldn't act on the $2$-d space of sections. I'm a little confused about extending that argument to the maximal torus, though. | |
| May 10, 2017 at 21:55 | history | edited | aglearner | CC BY-SA 3.0 |
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| May 10, 2017 at 11:09 | history | asked | aglearner | CC BY-SA 3.0 |